Descriptive evidence about firm death and dividends
A toy model that explains some of this evidence
(If time permits) a connection to the equity volatility puzzle
How I am working with firm death:
Given the pricing kernel, \(M\), and payoffs, \(X\), when would
\[ P = E\left(MX\right) = 0 \]
given perfect knowledge of firm cashflows.
| (1) | (2) | |
| Dividend Growth | -0.391*** | -0.038*** |
| (0.002) | (0.004) | |
| Cut to Zero | 4.749*** | |
| (0.029) | ||
| Observations | 765,795 | 765,795 |
| R2 | 0.022 | 0.055 |
| Max. Possible R2 | 0.262 | 0.262 |
| Note: | *p<0.1; **p<0.05; ***p<0.01 | |
Three periods, \(t \in \{0, 1, 2\}\)
Two types, \(v \in \{L, H\}\).
Let \(s_{v, t}\) denote the share of firms of type \(v\) at time \(t\).
Firms have earnings, \(e_t\) at time \(t \in \{1, 2\}\), with
\[ e_t = \begin{cases} \bar{e} \ \text{with probability } \lambda_v \\ \underline{e} \ \text{with probability } (1 - \lambda_v) \end{cases} \]
To alleviate incentive problems, managers are paid a portion of the stock price at periods \(t \in \{0, 1\}\), where
\[ S_0 = E^M_t\left(d_1 + d_2\right)\\ S_1 = E^M_t\left(d_2\right)\\ S_2 = 0 \]
Managers have wage \(w_t = \alpha S_t\) for some \(\alpha \in (0, 1)\). Managers know their types, but the market does not.
In periods \(t \in \{0, 1\}\) managers commit to a dividend policy which is a function \(d_t: e_{t+1} \to \mathbb{R}\).
This is subject to the constraint
\[ \Delta c_t + d_t \leq e_t \]
with the additional stipulation that if \(c_1 = 0\) the firm dies in that period and cannot receive earnings in period 2.
Note that in any equilibrium, managers commit to liquidating the firm at the end of period 2.
There are two equilibria of interest here.
Strategy profiles:
High types, \(v = {H}\)
\[d_{1,H} = \begin{cases} \underline{e} \text{ when } e_1 = \underline{e} \\ 0 \lt d_1 \lt \bar{e} \text{ otherwise} \end{cases}\]
Types \(H\) commit to ritual suicide in period one in order to signal their type distribution of firms in period two.
Low types, \(v = {L}\)
\[d_{1,L} = 0 \leq d_1 \lt e_1\]
Since there is no discounting, as long as they survive to period 2 their valuation is the same.
High types don’t deviate if
\[S_{1, H} + \lambda_H S_{1, H} - 2S_{1, L} > 0\]
Low types don’t deviate if
\[2S_{1, L} - S_{1, H} - \lambda_L S_{1, H} > 0\]
Let’s start with Shiller’s constant discount rate model:
\[ S_{i,t} = E^M\left(\sum_{s=t}^\infty \frac{d_sP(\text{alive})}{(1 + r)^t} \right) \]
Suppose we have a survival rate, \(\lambda\), where dividends contain information about \(\lambda\). Then
\[ S_{i,t} = E^M\left(\sum_{s=t}^\infty \frac{d_s \lambda(d_{i, 0:t})^t}{(1 + r)^t} \right) \]
If we assume a constant future dividend (just for simplicity), we get
\[ S_{i, t} = \frac{d_{i,t}}{r - (\lambda(d_{i,0:t}) - 1)} = \frac{d_{i,t}}{r + (1 - \lambda(d_{i,0:t}))} \]
which implies that changes in the hazard rate look like changes in the discount rate.
At the individual firm level, dividends are very informative about the path of future cashflows
This pattern can be rationalized with a signaling model
It seems like this effect is not large enough to generate the variation in equity prices that we would need
Next steps would be to make the model dynamic / have a continuum of firms / have a continuum of payoffs (maybe in continuous time?)
I model the information that dividends contain about firm death with a Cox proportional hazard model.
This model is semiparametric because we can divide out the baseline hazard. For two observations with different dividend changes,
\[\frac{\lambda(t | \Delta d_i)}{\lambda(t | \Delta d_j)} = \exp(\beta (\Delta d_i - \Delta d_j))\]
so we don’t need to model the baseline death rate.
Period 0:
\[ \begin{aligned} S_{0, H} = \lambda_H \bar{e} + (1 - \lambda_H )\underline{e} + \lambda_H(\lambda_H \bar{e} + (1 - \lambda_H)\underline{e}) \end{aligned} \]
\[ \begin{aligned} S_{0, L} = 2(\lambda_L \bar{e} + (1 - \lambda_L )\underline{e}) \end{aligned} \]
Period 1:
\[ \begin{aligned} S_{1, H} = \lambda_H \bar{e} + (1 - \lambda_H )\underline{e} \end{aligned} \]
\[ \begin{aligned} S_{1, L} = \lambda_L \bar{e} + (1 - \lambda_L )\underline{e} \end{aligned} \]
We have many MM equilibria, where the level of dividends in period 1 is irrelevant to prices, and all firms choose the same strategy.
A MM equilibrium has the following strategies:
For types \(v \in \{H, L\}\).
\[0 \leq d_{1,v} \lt e_1\]
The market updates their beliefs given observed earnings via Bayes’ rule.
\[P_0(v = v') = s_{v', 0}\] \[P_1(v = v' | e = \bar{e}) = \frac{s_{v', 0} \lambda_{v'}}{s_{v', 0}\lambda_{v'} + s_{v'', 0}\lambda_{v''}}\]
\[P_1(v = v' | e = \underline{e}) = \frac{s_{v', 0} (1 - \lambda_H)}{s_{{v'}, 0}(1 - \lambda_{v'}) + s_{{v''}, 0}(1 - \lambda_{v''})}\]
Initial prices: \[S_0 = 2 \left[s_{H, 0} (\lambda_H \bar{e} + (1 - \lambda_H)\underline{e}) + s_{L, 0} (\lambda_L \bar{e} + (1 - \lambda_L)\underline{e}) \right]\]
Prices conditional on high earnings: \[S_{1|\bar{e}_1} = \frac{s_{H, 0} \lambda_H (\lambda_H \bar{e} + (1 - \lambda_H)\underline{e}) + s_{L, 0} \lambda_L (\lambda_L \bar{e} + (1 - \lambda_L)\underline{e})}{s_{H, 0} \lambda_H + s_{L, 0} \lambda_L}\]
Prices conditional on low earnings: \[S_{1|\underline{e}_1} = \frac{ \Sigma_{v \in\{H, L\}} (s_{v, 0} (1 - \lambda_v) (\lambda_v \bar{e} + (1 - \lambda_v)\underline{e})}{s_{v, 0} (1 - \lambda_H) + s_{L, 0} (1 - \lambda_L)}\]
It is easy to sustain these kind of equilibria with the right market punishments. A standard refinement in these games is the intuitive criterion (Cho and Krebs, 1987).
Let \(S_{0, H}\), \(S_{1, H}\), \(S_{0, L}\), \(S_{1, L}\) be the stock prices of from the separating equilibrium.
For this to satisfy the intuitive criterion,
\[ \begin{aligned} (1 + \lambda_h - 2 s_{H, 0})E(e_t | v = H) \leq s_{L, 0} S_{L, 0} - (S_{H, 1} - E_0(S_1 | v = H)) \end{aligned} \]
A bit ugly but it makes sense.